106       V   NORMED SPACES
 (a)	Show that if/satisfies condition (1) and is continuous at the point 0, it is con-
tinuous in a neighborhood of 0 and is linear in a neighborhood of 0 (cf. Problem 1).
 (b)	Let g be a mapping of E into F; in order that g be continuous at the point 0 and
linear in a neighborhood of 0, a necessary and sufficient condition is that for any
convergent series (xn) in E, the partial sums of the series (g(xn) be bounded in F.
(To prove sufficiency, first observe that one must have #(0) = 0; if, for every n, there
exist three elements u»,vn,wn of E such that  \\un\\ < 2~", ||oj| < 2~", Hw.ll ^2-",
un + vn 4- m>b = 0 and#(wn) + g(vn) +g(wn) ^ 0, form a series (xn) violating the assump-
tion. If there are no such sequences un, vn, wn, g verifies condition (1); show that it is
necessarily continuous at 0.)
6. EQUIVALENT NORMS
Let E be a vector space (over the real or the complex field), \\x\\i and
||a:||2 two norms on E; we say that \\x\\i is finer than ||#||2 if the topology
defined by \\x\\i is finer than the topology defined by ||jc||2 (see Section 3.12);
if we note El (resp. E2) the normed space determined by \\x\\i (resp. ||jc||2),
this means that the identity mapping x -> x of e! into E2 is continuous,
hence, by (5.5.1), that condition is equivalent to the existence of a number
a>0 such that ||x||2 < a- \\x\\l9 We say that the two norms \\x\\l9 \\x\\2
are equivalent if they define the same topology on E. The preceding remark
yields at once:
(5.6.1) In order that the two norms \\x\\i, \\x\\ 2ona vector space E be equivalent
a necessary and sufficient condition is that there exist two constants a > 0,
i>0, such that
«Wi<IMl2<6|l*lli
for any xeE.
The corresponding distances are then uniformly equivalent (Section 3.14).
For instance, on the product Er x E2 of two normed spaces, the norms
sup(KIUM), \M\ + \\x2\\, (\\xtf + \\x2\\2)1'2 are equivalent. On the
space E = ^R(I), the norm H/ld defined in (5.1.4) is not equivalent to the
norm \\f\\n = sup|/(f)| (see Section 5.1, Problem 1).
tel
7. SPACES OF CONTINUOUS MULTILINEAR MAPPINGS
Let E, F, be two normed spaces; the set jSf(E;F) of all continuous
linear mappings of E into F is a vector space, as follows from (5.1.5), (3.20.4),
and (3.11.5).